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Question #40981

Difficulty: 8/10 | 🟢 Free

The Question

A subset $A$ of 4 distinct numbers is chosen uniformly at random from all subsets of $S = \{1, 2, 3, \dots, 15\}$ that have the property that no two elements in the subset sum to 15. Let $M$ be the expected value of the arithmetic mean of the elements in $A$ . Find the value of $M$ .

Options

A.
8
B.
$\frac{65}{8}$
C.
$\frac{33}{4}$
D.
$\frac{127}{16}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Properties of Arithmetic Mean|Sampling With/Without Replacement|Symmetry and Invariance

Why This Question Is Tricky

Common Mistakes

This question tests the distinction between the population mean and the expected value of a sample mean under biased selection constraints.

Students often fall into the 'Symmetry and Invariance' trap, assuming that the expected value of the sample mean is always equal to the population mean ( $\mu = 8$ ), relying on the linearity of expectation.

However, the restriction 'no two elements sum to 15' breaks the uniform symmetry of the sample space.

Because the element '15' is unpaired, it appears in a disproportionately high number of valid subsets relative to its value, shifting the expected mean upwards.

The trap punishes those who blindly apply $E[\bar{X}] = \mu$ without analyzing the restricted probability distribution.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Statistics - Class 11
Concept	Properties of Arithmetic Mean
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free